# Is there any algebraic structure in the set of all continuous pdf's in $\mathbb{R}^n$ under some operation?

It is possible to define an algebraic structure to the set of all continuous probability densities under certain operation ?

Example: Let $D = \{f(x_1,...,x_n) \mbox{ | } \int f(x_1,...,x_n)dx_1,...,dx_n = 1 \}$

This set posses any algebraic structure under certain operations such as multiplication, division, composition or any other special operation ?

One way you can do it is by treating $f_{X}(x)=f(x)$ as the pdf of a random variable. Then as others pointed out in the comments, $Z=X+Y$ is essentially defined by convolution of $f_{X}, f_{Y}$. The role of the identity should be given by the pointed measure $\delta_{0}$. But mind this is not a a continuous pdf anymore.
Similarly you can define the random variable $XY$ and try to compute $f_{XY}$ using so called "Jacobian method". It is easy to see that under this operation you get a commutative ring with is far from an integral domain. You may want to introduce division into the picture by using distributions; but it complicates the theory considerably. In general the calculus of distributions is not easy to compute in practice even without the constraint of being probability distributions.